In my differential equations course we are looking at systems of DDEs and I was wondering if there is a particular method that is used to solve these types of systems? One example that was given in class is the following; $$\begin{align}v_1^\prime(t)&=\phi(t)v_1(t)\\ v_2^\prime(t)&=\alpha(v_1(t-1)-v_2(t-1)) \end{align}$$ Where $\alpha\in\mathbb{R}$ and $\phi(t)$ is some quasi-periodic function. Does an analytical solution exist? Or should I relay on numerical solutions as I have got here
Where x1 and x2 are v1 and v2 respectively, solutions to the above equation with $\phi(t)=1+2(t+\omega) (mod\ 1)$ for $w$ random.
From the first ODE we have
$$ v_1(t) = C_0 e^{\int_0^t \phi(\tau)d\tau} $$
and the second after substitution
$$ v_2'(t) + \alpha v_2(t-1) = C_0 e^{\int_0^{t-1}\phi(\tau)d\tau} $$
to solve this ODE you can use many numerical methods and also when $\phi(t)$ is a linear function like $b t$, the Laplace transform can provide algebraically suitable methods to make parametric considerations, as follows
$$ \alpha \left(e^{-s} \left(\mathcal{L}_t[v_2(t)](s)\right)+e^{-s} \int_{-1}^0 e^{-s t} v_2(t) \, dt\right)+s \left(\mathcal{L}_t[v_2(t)](s)\right)-\frac{\sqrt{\pi } C_0 e^{\frac{(c-s)^2}{4 b}} \left(\text{erf}\left(\frac{c-s}{2 \sqrt{b}}\right)+1\right)}{2 \sqrt{b}}-v_2(0)=0 $$
or calling $V_2(s) = \mathcal{L}_t[v_2(t)](s)$
$$ V_2(s) = \frac{\sqrt{\pi } C_0 e^{\frac{(c-s)^2}{4 b}+s} \left(\text{erf}\left(\frac{c-s}{2 \sqrt{b}}\right)+1\right)-2 \sqrt{b} \left(\alpha \int_{-1}^0 e^{-s t} v_2(t) \, dt-e^s v_2(0)\right)}{2 \sqrt{b} \left(\alpha +e^s s\right)} $$
now making some assumptions like $v_2(t) = 0$ for $-1\le t\le 0$ and using for $e^s$ the Padé expansion we can proceed making stability considerations involving $\{\alpha, b, c\}$ as
$$ V_2(s) = \frac{\sqrt{\pi } C_0e^{\frac{(c-s)^2}{4 b}+s} \left(\text{erf}\left(\frac{c-s}{2 \sqrt{b}}\right)+1\right)}{2 \sqrt{b} \left(\alpha +e^s s\right)} $$